Contact structures and Floer homology on 3-manifolds with boundary

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The principal investigator aims to understand the interplay between contact geometry in 3 dimensions and invariants of 3-manifolds. Contact structures are geometric objects which originated in the study of optics in the 19th century; in recent years they have become important tools in low-dimensional topology, providing information about the shapes of 3- and 4-dimensional spaces such as our universe, and in knot theory, which has seen emerging links to such topics as protein folding. Many of the tools which the PI uses to study such spaces, including Floer homology theories, draw heavily from mathematical physics, and they contain contact-geometric data which has provided a wealth of information about both the contact structures themselves and about the spaces in which they reside. The proposed research would investigate such data and uncover connections which it provides between disparate areas of mathematics, including algebra, topology, and dynamics.The PI intends to study invariants of contact structures and 3-manifolds, especially homological invariants of 3-manifolds with boundary, which arise from gauge theory and symplectic geometry. The first goal of this project is to develop applications to topology and to symplectic geometry of contact invariants in several Floer homology theories for sutured 3-manifolds. These potential applications include computable obstructions to Lagrangian concordances between Legendrian knots; bounds on the number of Reeb orbits in a contact 3-manifold, generalizing the proof of the Weinstein conjecture in this setting; and an intriguing conjecture relating Stein fillings of a 3-manifold to representations of its fundamental group. The second goal is to establish a relationship between different sutured Floer homology theories, whose corresponding closed 3-manifold invariants (Heegaard Floer homology, monopole Floer homology, and embedded contact homology) are now all known to be isomorphic, and to identify their respective contact invariants as well. The third goal is to investigate an emerging connection between Legendrian contact homology (LCH) and new sheaf-theoretic Legendrian knot invariants, and in doing so to apply algebro-geometric techniques to problems in contact geometry and hopefully understand the relationship of LCH to classical knot invariants.